Test Your Knowledge of Information Theory

By Jeffrey Shallit

Posted: January 11, 2009

Creationists think information theory poses a serious challenge to modern evolutionary biology -- but that only goes to show that creationists are as ignorant of information theory as they are of biology.

Whenever a creationist brings up this argument, insist that they answer the following five questions. All five questions are based on the Kolmogorov interpretation of information theory. I like this version of information theory because (a) it does not depend on any hypothesized probability distribution (a frequent refuge of scoundrels) (b) the answers about how information can change when a string is changed are unambiguous and agreed upon by all mathematicians, allowing less wiggle room to weasel out of the inevitable conclusions, and (c) it applies to discrete strings of symbols and hence corresponds well with DNA.

All five questions are completely elementary, and I ask these questions in an introduction to the theory of Kolmogorov information for undergraduates at Waterloo. My undergraduates can nearly always answer these questions correctly, but creationists usually cannot.

Q1: Can information be created by gene duplication or polyploidy? More specifically, if x is a string of symbols, is it possible for xx to contain more information than x?

Q2: Can information be created by point mutations? More specifically, if xay is a string of symbols, is it possible that xby contains significantly more information? Here a, b are distinct symbols, and x, y are strings.

Q3: Can information be created by deletion? More specifically, if xyz is a string of symbols, is it possible that xz contains signficantly more information?

Q4: Can information be created by random rearrangement? More specifically, if x is a string of symbols, is it possible that some permutation of x contains significantly more information?

Q5. Can information be created by recombination? More specifically, let x and y be strings of the same length, and let s(x, y) be any single string obtained by "shuffling" x and y together. Here I do not mean what is sometimes called "perfect shuffle", but rather a possibly imperfect shuffle where x and y both appear left-to-right in s(x, y) , but not necessarily contiguously. For example, a perfect shuffle of 0000 and 1111 gives 01010101, and one possible non-perfect shuffle of 0000 and 1111 is 01101100. Can an imperfect shuffle of two strings have more information than the sum of the information in each string?

The answer to each question is "yes". In fact, for questions Q2-Q5, I can even prove that the given transformation can arbitrarily increase the amount of information in the string, in the sense that there exist strings for which the given transformation increases the complexity by an arbitrarily large multiplicative factor. I won't give the proofs here, because that's part of the challenge: ask your creationist to provide a proof for each of Q1-Q5.

Now I asserted that creationists usually cannot answer these questions correctly, and here is some proof.

Q1. In his book No Free Lunch, William Dembski claimed (p. 129) that "there is no more information in two copies of Shakespeare's Hamlet than in a single copy. This is of course patently obvious, and any formal account of information had better agree." Too bad for him that Kolmogorov complexity is a formal account of information theory, and it does not agree.

Q2. Lee Spetner and the odious Ken Ham are fond of claiming that mutations cannot increase information. And this creationist web page flatly claims that "No mutation has yet been found that increased the genetic information." All of them are wrong in the Kolmogorov model of information.

Q4. R. L. Wysong, in his book The Creation-Evolution Controversy, claimed (p. 109) that "random rearrangements in DNA would result in loss of DNA information". Wrong in the Kolmogorov model.

So, the next time you hear these bogus claims, point them to my challenge, and let the weaselling begin!